An Elliptic Neumann Problem Involving Critical Exponent

This paper is devoted to study the following nonlinear elliptic problem with Neumann boundary condition, (Pμ) : −Δu + μu = Ku3 , u > 0 in Ω and ∂u/∂v = 0 on ∂Ω where Ω is a smooth bounded domain in R4, μ is a positive parameter and K is a C3 positive Morse function on Ω. Using dynamical methods involving the study of Palais-Smale condition of the associated variational structure J, we prove some existence results of (P μ).